Theorem 1.3 
If \( \alpha \in \mathrm {DC}_d\) , \( v\in G_{r_0}^\nu (d,{\mathbb {R}}) \) , then for any \( r\in (0,r_0) \) , there exists \( \lambda _0(\alpha ,d,\nu ,r_0,r) \) , such that if \( \lambda >\lambda _0 \) , \( L_{\lambda ^{-1} v,\alpha ,\phi } \)  has pure point spectrum with sub-exponentially decaying eigenfunctions for a.e. \( \phi \in .\) 
If \( d=1 \) , Bourgain-Jitomirskaya [1]} showed that for any fixed Diophantine \( \alpha \) , \( L_{\lambda ^{-1} v,\alpha ,\theta } \)  has Anderson localization for sufficiently large \( \lambda \)  and a.e. \( \theta \) . This result is non-perturbative in the sense that the largeness of \( \lambda \)  is independent of \( \alpha \) . Recently, Ge-You-Zhou [2]} showed the same holds for \( d\ge 2 \) , but is a perturbative result, in the sense that the largeness of \( \lambda \)  depends on \( \alpha \) . Theorem REF  extends their result to Gevrey class. As a comparison, we mention that Shi [3]} proved that for multi-frequency quasi-periodic operator with Gevrey type perturbations, that is,
\((L_{\lambda , v,f,\alpha ,\phi }u)_n=\sum _{k\in {\mathbb {Z}}^d}\hat{v}(n-k)u_k+\lambda f(\phi +\langle n,\alpha \rangle )u_n, \phi \in d,\) 
